Contributions à l'Etude du Vertex Topologique en Théorie ... - Toubkal

L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341 319〈Π ia | stands for the dual state associated with the dual partition Π + = ˜Π T .Wealsohavethefollowing relationI id =∑|Π ia 〉〈Π ia |, 〈Π ia |Π jb 〉=δ ij δ ab ,(3.30)3d partitionsdefining the resolution of the identity operator I id .(iv) The number |Π| of unit boxes (cubes) of the 3d partition is defined as|Π|= ∑ i,aΠ i,a .(3.31)(v) The boundary (∂Π) of the 3d partition Π is given by the 2d profile of the correspondinggeneralized Young diagram. As this property is important for the present study, let give somedetails.Given a 3d partition Π,theboundary term on the plane x i = N i is a Young diagram (2d partition).On the planes x 1 = N 1 , x 2 = N 2 and x 3 = N 3 , the boundary of Π is composed of by three 2dpartitions λ, μ and ν. So we then have:∂Π = (λ,μ,ν).(3.32)Particular boundaries are given by the case where a 2d partition is located at infinity; that is thereis no boundary. We distinguish the following situations:∂Π = (∅,μ,ν),∂Π = (∅, ∅,ν),∂Π = (∅, ∅, ∅),where ∅ stands for the vacuum.(3.33)(vi) A convenient way to deal with 3d partitions is to slice them as a sequence of 2d partitionswith interlacing relations. We mainly distinguish two kinds of sequences of 2d partitions:perpendicular and diagonal. We will not need this property here; but for details on this mattersee for instance [30] and references therein.β) 4d partitions The 4d partitions P are extensions of the 3d partitions Π considered above.They can be imagined as 4d generalized Young diagrams described by the typical integral rank3-tensor,P iaα ∈ Z + , with P iaα P (i+j)(a+b)(α+β) ,(3.34)with 1 i N 1 ,1 a N 2 and 1 α N 3 .Several properties of 2d and 3d partitions extend to the 4d case; there are also specific propertiesin particular those concerning their slicing into lower-dimensional ones. Below we describesome particular properties of 4d partitions by considering special representations.Sub-classes of 4d partitions are given by:(i) the product of a 2d and a 3d partitions μ and Π likeP = μ ⊗ Π, (P iaα ) = (μ i Π aα ),(3.35)